arXiv subjects

O. C. Martin

Publications and source records attributed to O. C. Martin.

43 records · Page 3Linked to original sources

arXiv1998

Comparing Mean Field and Euclidean Matching Problems

Combinatorial optimization is a fertile testing ground for statistical physics methods developed in the context of disordered systems, allowing one to confront theoretical mean field predictions with actual properties of finite dimensional systems. Our focus here is on minimum matching problems, because they are computationally tractable while both frustrated and disordered. We first study a mean field model taking the link lengths between points to be independent random variables. For this model we find perfect agreement with the results of a replica calculation. Then we study the case where the points to be matched are placed at random in a d-dimensional Euclidean space. Using the mean field model as an approximation to the Euclidean case, we show numerically that the mean field predictions are very accurate even at low dimension, and that the error due to the approximation is O(1/d^2). Furthermore, it is possible to improve upon this approximation by including the effects of Euclidean correlations among k link lengths. Using k=3 (3-link correlations such as the triangle inequality), the resulting errors in the energy density are already less than 0.5% at d>=2. However, we argue that the Euclidean model's 1/d series expansion is beyond all orders in k of the expansion in k-link correlations.

cond-mat.dis-nn↗

arXiv1998

Ising Spin Glasses in a Magnetic Field

Ground states of the three dimensional Edwards-Anderson spin glass are computed in the presence of an external magnetic field. Our algorithm is sufficiently powerful for us to treat systems with up to 600 spins. We perform a statistical analysis of how the ground state changes as the field is increased, and reach the conclusion that the spin glass phase at zero temperature does not survive in the presence of any finite field. This is in agreement with the droplet model or scaling predictions, but in sharp disagreement with the mean field picture. For comparison, we also investigate a dilute mean field spin glass model where an Almeida-Thouless line is present.

cond-mat.dis-nn↗

arXiv1998

The stochastic traveling salesman problem: Finite size scaling and the cavity prediction

We study the random link traveling salesman problem, where lengths l_ij between city i and city j are taken to be independent, identically distributed random variables. We discuss a theoretical approach, the cavity method, that has been proposed for finding the optimal tour length over this random ensemble, given the assumption of replica symmetry. Using finite size scaling and a renormalized model, we test the cavity predictions against the results of simulations, and find excellent agreement over a range of distributions. We thus provide numerical evidence that the replica symmetric solution to this problem is the correct one. Finally, we note a surprising result concerning the distribution of kth-nearest neighbor links in optimal tours, and invite a theoretical understanding of this phenomenon.

cond-mat.dis-nn↗

arXiv1998

Droplet Phenomenology and Mean Field in a Frustrated and Disordered System

The low lying excited states of the three-dimensional minimum matching problem are studied numerically. The excitations' energies grow with their size and confirm the droplet picture. However, some low energy, infinite size excitations create multiple valleys in the energy landscape. These states violate the droplet scaling ansatz, and are consistent with mean field predictions. A similar picture may apply to spin glasses whereby the droplet picture describes the physics at small length scales, while mean field describes that at large length scales.

cond-mat.dis-nn↗

arXiv1998

Cut Size Statistics of Graph Bisection Heuristics

We investigate the statistical properties of cut sizes generated by heuristic algorithms which solve approximately the graph bisection problem. On an ensemble of sparse random graphs, we find empirically that the distribution of the cut sizes found by ``local'' algorithms becomes peaked as the number of vertices in the graphs becomes large. Evidence is given that this distribution tends towards a Gaussian whose mean and variance scales linearly with the number of vertices of the graphs. Given the distribution of cut sizes associated with each heuristic, we provide a ranking procedure which takes into account both the quality of the solutions and the speed of the algorithms. This procedure is demonstrated for a selection of local graph bisection heuristics.

cond-mat.stat-mech↗

arXiv1998

The random link approximation for the Euclidean traveling salesman problem

The traveling salesman problem (TSP) consists of finding the length of the shortest closed tour visiting N ``cities''. We consider the Euclidean TSP where the cities are distributed randomly and independently in a d-dimensional unit hypercube. Working with periodic boundary conditions and inspired by a remarkable universality in the kth nearest neighbor distribution, we find for the average optimum tour length = beta_E(d) N^{1-1/d} [1+O(1/N)] with beta_E(2) = 0.7120 +- 0.0002 and beta_E(3) = 0.6979 +- 0.0002. We then derive analytical predictions for these quantities using the random link approximation, where the lengths between cities are taken as independent random variables. From the ``cavity'' equations developed by Krauth, Mezard and Parisi, we calculate the associated random link values beta_RL(d). For d=1,2,3, numerical results show that the random link approximation is a good one, with a discrepancy of less than 2.1% between beta_E(d) and beta_RL(d). For large d, we argue that the approximation is exact up to O(1/d^2) and give a conjecture for beta_E(d), in terms of a power series in 1/d, specifying both leading and subleading coefficients.

cond-mat↗

arXiv1998

Scaling universalities of kth-nearest neighbor distances on closed manifolds

Take N sites distributed randomly and uniformly on a smooth closed surface. We express the expected distance from an arbitrary point on the surface to its kth-nearest neighboring site, in terms of the function A(l) giving the area of a disc of radius l about that point. We then find two universalities. First, for a flat surface, where A(l)=πl^2, the k-dependence and the N-dependence separate in . All kth-nearest neighbor distances thus have the same scaling law in N. Second, for a curved surface, the average \int dμover the surface is a topological invariant at leading and subleading order in a large N expansion. The 1/N scaling series then depends, up through O(1/N), only on the surface's topology and not on its precise shape. We discuss the case of higher dimensions (d>2), and also interpret our results using Regge calculus.

math.DG↗